Nonlinear Wave Research Articles

Towards efficient control synthesis for nonlinear wave energy conversion systems: impedance-matching meets the spectral-domain

AbstractExisting studies within the literature that focus on designing parametric energy-maximizing controllers for Wave Energy Converter (WEC) systems predominantly rely on the impedance-matching (IM) principle, originally developed for linear time-invariant systems. Alternatively, iterative optimization routines are commonly employed for nonlinear WECs. However, these approaches often face a trade-off between effectiveness in maximizing energy extraction and computational efficiency. To address this limitation, this study proposes a computationally efficient controller tuning method for analogous synthesis in the case of nonlinear WECs. The proposed approach combines a statistical linearization technique known as spectral-domain modeling with the IM principle, to synthesize a Proportional–Integrative (PI) controller for a nonlinear WEC. Furthermore, a comparison is performed with two other synthesis methods: one based on a standard (i.e. linear) frequency-domain representation of the WEC that incorporates the IM principle, and the other employing a gradient-free optimization routine applied to the nonlinear time-domain model of the WEC for PI parameter tuning through exhaustive numerical search. A discussion on the effectiveness of each tuning method in maximizing energy absorption is provided, including an appraisal of their associated computational time requirements. Numerical analyses demonstrate that the proposed method, which integrates spectral-domain modeling and IM, can achieve (almost) optimal PI controller design for a nonlinear WEC. Furthermore, this study addresses the inaccuracies inherent in the frequency-domain approach and significantly reduces the computational time compared to the exhaustive search procedure. The findings of this research represent a significant advancement towards the development of simple, effective, and efficient IM-based techniques for synthesis of controllers in nonlinear WEC systems

Exploring the physical characteristics and nonlinear wave dynamics of a (3+1)-dimensional integrable evolution system

This study comprehensively explores the [Formula: see text]-dimensional Mikhailov–Novikov–Wang [Formula: see text] integrable equation, with the primary objective of elucidating its physical manifestations and establishing connections with analogous nonlinear evolution equations. The investigated model holds significant physical meaning across various disciplines within mathematical physics. Primarily, it serves as a fundamental model for understanding nonlinear wave propagation phenomena, offering insights into wave behaviors in complex media. Moreover, its relevance extends to nonlinear optics, where it governs the dynamics of optical pulses and solitons crucial for optical communication and signal processing technologies. Employing analytical methodologies, namely the unified [Formula: see text], Khater II ([Formula: see text]hat.II) method, and He’s variational iteration [Formula: see text] method, both numerical and analytical solutions are meticulously examined. Through this investigation, the intricate behaviors of the equation are systematically unveiled, shedding illuminating insights on various physical phenomena, notably including wave propagation in complex media and nonlinear optics. The outcomes not only underscore the efficacy of the analytical techniques deployed but also accentuate the equation’s pivotal role in modeling a broad spectrum of nonlinear wave dynamics. Consequently, this research significantly advances our comprehension of complex physical systems governed by nonlinear dynamics, thereby contributing notably to interdisciplinary pursuits in mathematical physics.

Applications of nonlinear waves; lump wave, rogue wave, periodic wave, breather wave for a nonlinear model

Applications of nonlinear waves; lump wave, rogue wave, periodic wave, breather wave for a nonlinear model

New solitary waves in a convecting fluid

It is critical to examine the effects of small perturbations on solitary wave solutions to nonlinear wave equations because of the existence of the unavoidable perturbations in real applications. In this work, we aim to study the solitary wave solutions for the dissipative perturbed mKdV equation which is proposed to model the evolution of shallow wave in a convecting fluid. By using the geometric singular perturbation theorem, the three-dimensional traveling wave system is reduced to a planar dynamical system with regular perturbation. By examining the homoclinic bifurcations via Melnikov method, we show that not only some solitary waves with particularly chosen wave speed persist but also some new type of solitary waves appear under small perturbation. The theoretical analysis results are illustrated by numerical simulations.

A Set of Accurate Dispersive Nonlinear Wave Equations

In this study, a set of accurate dispersive nonlinear wave equations is established, using the wave velocity and free surface elevation as variables. These equations improve upon previous equations in which the velocity potential is used as a variable by considering the rotational wave motion and by adding a second-order bottom slope term that applies to general situations, allowing the equations to consider the influence of rapidly changing, horizontal, two-dimensional bottom topographies. The problem of the inaccuracy of the integral calculations used in previous equations in nearshore areas is solved by approximating the integral calculations into differential calculations, and a set of coupled wave equations is established by keeping the free surface elevation and the horizontal velocity constant, thus allowing the calculation of nearshore wave-generated currents. The benefits of the current model are confirmed through comparisons with corresponding laboratory experimental findings and are illustrated through a comparison with the numerical outcomes of other pertinent models.

Measurement Of Spatial-Temporal Waves In The Laboratory Using Computer Vision Technolog

Given the complexity of ocean wave phenomena, physical modeling continues to be essential in various engineering and research applications. Wave height, a characteristic measured in nearly all projects, is commonly assessed using resistance/capacitance wave gauges. However, a significant limitation of these traditional wave gauges is their point-based nature, which restricts their ability to describe detailed spatial wave characteristics, especially during wave nonlinearity or when waveforms change. Furthermore, to reconstruct the 3D wave field, an array of wave gauges is often employed alongside an appropriate interpolation algorithm. Additionally, wave gauges may alter flow/wave conditions in specific scenarios. To address these limitations, various non-intrusive measurement techniques have been proposed. Gomit et al. (2022) classified these into stereoscopic, projection, and light-based methods, including LiDAR (Blenkinsopp et al., 2012), RGB-D cameras (Martínez-Aranda et al., 2018), and binocular stereo vision (Li et al., 2022). However, their applicability is constrained by factors such as optical setup complexity and device cost. With the rapid advancement of computer vision technology, it is possible to improve these issues and better describe temporal and spatial wave field variations using such techniques. This conference contribution details the reconstruction of the temporal and spatial 3D free water surface using photogrammetry techniques applied to images from consumer webcams. It also compares these results with measurements obtained from a wave gauge.

Transcending classical diffusion models: nonlinear dynamics and solitary waves in the fractional Chaffee–Infante equation

Transcending classical diffusion models: nonlinear dynamics and solitary waves in the fractional Chaffee–Infante equation

Auto-Bäcklund Transformation with the Solitons and Similarity Reductions for a Generalized Nonlinear Shallow Water Wave Equation

Auto-Bäcklund Transformation with the Solitons and Similarity Reductions for a Generalized Nonlinear Shallow Water Wave Equation

Performance Assessment Of Two Active Absorption Systems In A Large Wave Flume

The quality of most coastal or ocean laboratory experiment depends on the accuracy, uniformity and stability of the generated waves. Generation of waves in the laboratory seems simple, but it should consider the spatio-temporal variability of the wave profile. As they propagate, waves interact with the boundaries, i.e. changes in water depth, lateral walls, test specimens, instrument supporting structures, passive absorbers, etc., as well as with varying hydrodynamic conditions (wave-induced currents, wave-wave interactions, or nonlinear energy transfer, among others). Reflected, scattered and radiated waves may propagate back to the wave generator and reflect back. Re-reflections from the wave generator may be a non-realistic representation of the intended wave field, to the extent that the total energy content in the testing facility may increase significantly and uncontrollably. Theoretically, for weakly nonlinear unidirectional waves propagating over a relatively small number of wavelengths, and as long as the still water depth and facility width does not change, the wave properties should remain constant in space and time, which can be identified as the uniformity and stability of the waves. Under these conditions, nonuniform or unsteady wave properties can be associated to an inadequate selection of the wave theory (Airy, Stokes n-order, Cnoidal, Stream Function, etc., e.g. Fenton 1999), the approximation of the wave generation technique (first or second order, nonlinear - see e.g. Mohtat et al., 2020), formation of wave instabilities (e.g. Benjamin-Feir, 1967), wave reflection, scattering or radiation, and the performance of the active wave absorption system. It can be said that the performance of the active wave absorption control system is one of the responsible factors that can affect directly the quality of the generated waves. Many laboratories have implemented different passive and active wave absorption systems, but the performance on the uniformity and stability of the generated waves, as well as the performance of the active wave absorption system used, have not been assessed in a systematic way, and related published data and information is quite scarce, with few interesting exceptions, e.g. (Lykke-Andersen, 2016; Spinneken 2010; Schäffer and Skourup 1996). However, a direct comparison among the different available active wave absorption control systems is still missing. This paper presents a thorough experimental study on the performance assessment of two different active wave absorption control systems (MTS and Awasys7) as a means to establish the quality of the experiment in terms of the uniformity (space) and stability (time) of generated wave parameters (descriptors) in the laboratory.

A 3-D minimum-enstrophy vortex in stratified quasi-geostrophic flows

Applying a variational analysis, a minimum-enstrophy vortex in three-dimensional (3-D) fluids with continuous stratification is found, under the quasi-geostrophic hypothesis. The buoyancy frequency is held constant. This vortex is an ideal limiting state in a flow with an enstrophy decay while energy and generalized angular momentum remain fixed. The variational method used to obtain two-dimensional (2-D) minimum-enstrophy vortices is applied here to 3-D integral quantities. The solution from the first-order variation is expanded on a basis of orthogonal spherical Bessel functions. By computing second-order variations, the solution is found to be a true minimum in enstrophy. This solution is weakly unstable when inserted in a numerical code of the quasi-geostrophic equations. After a stage of linear instability, nonlinear wave interaction leads to the reorganization of this vortex into a tripolar vortex. Further work will relate our solution with maximal entropy 3-D vortices.

Diverse interference of nonlinear waves on the non-zero background in the coherently coupled system

Diverse interference of nonlinear waves on the non-zero background in the coherently coupled system

Validation Of A Nonlinear Wave Decompistion Method Including Shoaling

It is important to decompose the incident and the reflected waves when performing physical or numerical experiments in a wave flume. Especially when large reflection is expected, from for example a breakwater, the total measured signal can significantly deviate from the incident signal. Different techniques exist to decompose a signal into incident and reflected signals. For 1D wave flumes a method based on co-located wave gauges (Nwogu, 1989) or multiple wave gauges (Røge Eldrup and Lykke Andersen, 2019, Zelt and Skjelbreia, 1993) is commonly applied. The latter is the focus of this abstract since recently, the decomposition method based on multiple wave gauges was extended to be applicable to nonlinear irregular waves by including bound waves and amplitude dispersion (Eldrup and Lykke Andersen, 2019). Moreover, the practical requirements for the nonlinear wave decomposition methods were described in De Ridder et al. (2023). Most of the nonlinear wave decomposition methods are only applicable to a flat bed and will introduce an error when it is applied on a sloping foreshore which is typically the case in physical model experiments. Padilla and Alsina (2020) derived a general framework including shoaling of bound waves and Lykke Andersen and Eldrup (2021) presented a method for nonlinear regular waves over a sloping bed. However, a nonlinear decomposition method for irregular waves over a sloping bed has never been verified with physical model experiments. In this abstract the nonlinear decomposition method for irregular waves as presented in De Ridder et al. (2023) is extended with the effects of shoaling and its effects are verified with a physical model experiment.

Several nonlinear waves in spin quantum electron-positron plasmas

Abstract By using the reductive perturbation method, we obtained a nonlinear Schrödinger equation considering spin properties for a magnetized electron-positron plasmas. Several nonlinear wave were studied. The results indicate that various types of nonlinear waves exist in a magnetized electron-positron plasmas and they are spatially localized in both parallel and vertical directions to the external magnetized field direction. Additionally, the dependence of the wave amplitudes, wave width in both direction, group velocity of the envelop wave and the phase velocity of the background waves for these kinds of the nonlinear waves on the system parameters are all given in the present paper.

Evolution of Subsonic Shock Waves Associated with Reconnection Jets in Earth’s Magnetotail

Motivated by the signatures of nonlinear electrostatic waves observed by the Magnetospheric Multiscale spacecraft mission in reconnection jet regions of Earth's magnetotail, we have explored the dynamical features of ion-acoustic shock waves in the magnetotail. In this investigation, we have examined the dynamics and characteristics of ion-acoustic subsonic shock waves in non-Maxwellian space plasma comprising of two counterstreaming ion beams with suprathermal electrons, assumed to follow a kappa (κ) distribution. A reductive perturbation technique has been adopted to establish an evolution equation for small amplitude electrostatic shock structures. Importantly, subsonic waves only exist when the beam velocity exceeds a certain threshold, beyond which supersonic and subsonic waves may coexist. The combined effects of the beam velocity and the non-Maxwellian electron statistics have been analyzed to examine the characteristics of subsonic shock waves. Both symmetric and asymmetric (in relative beam density) models have been considered, leading to distinct possibilities in the evolution of subsonic shock waves. The findings of the investigation will help unfold the relatively unexplored dynamical characteristics of subsonic shock waves that may form and propagate in the magnetosphere.

Nonlinear gravitational waves in Horndeski gravity: scalar pulse and memories

We present and analyze a new non-perturbative radiative solution of Horndeski gravity. This exact solution is constructed by a disformal mapping of a seed solution of the shift-symmetric Einstein-Scalar system belonging to the Robinson-Trautman geometry describing the gravitational radiation emitted by a time-dependent scalar monopole. After analyzing in detail the properties of the seed, we show that while the general relativity solution allows for shear-free parallel transported null frames, the disformed solution can only admit parallel transported null frames with a non-vanishing shear. This result shows that, at the nonlinear level, the scalar-tensor mixing descending from the higher-order terms in Horndeski dynamics can generate shear out of a pure scalar monopole. We further confirm this analysis by identifying the spin-0 and spin-2 polarizations in the disformed solution using the Penrose limit of our radiative solution. Finally, we compute the geodesic motion and the memory effects experienced by two null test particles with vanishing initial relative velocity after the passage of the pulse. This exact radiative solution offers a simple framework to witness nonlinear consequences of the scalar-tensor mixing in higher-order scalar-tensor theories.

Analysis of nonlinear wave propagation within architected materials consisting of nonlinear Timoshenko beam structural elements

In the present work, a full nonlinear Timoshenko beam employing nonlinear shape functions is developed. The extended Hamilton principle is employed for deriving the differential equations of motion and the associated boundary conditions. The general form of the boundary conditions is then utilized to determine the static solution of the beam motion. Using this solution for the deformation and rotation of the beam, the nonlinear shape functions of the beam are identified, which leads to the linear and nonlinear mass and stiffness matrices of the Timoshenko beam element. The nonlinear dispersion diagram incorporating the non-linear corrections is obtained using the Linstedt-Poincaré perturbation method. An analysis of the effect of internal transverse shear and bending on the nonlinear dispersion characteristics of wave propagation in two-dimensional periodic network materials made of nonlinear Timoshenko beams is done. The formulated theory shows that the percentage of correction factor of the nonlinear kinematics versus the linear dynamical behavior is inversely proportional to the frequency amplitude. The shear and extension modes are shown to have the higher effect in the non-linear correction term in comparison to the flexural mode.

Some novel different solutions for Boussinesq-type models including bright, singular, and dark soliton ones

Some new different kinds of one-soliton solutions for various forms of Boussinesq-type equations are presented in thispaper to describe the nonlinear wave phenomena in coastal and ocean areas such as tsunami waves. These one-solitonsolutions include bright, dark, and singular ones. The property of each solution in coastal and ocean engineering isexplained.

Bifurcation analysis, quasi-periodic and chaotic behavior of generalized Pochhammer-Chree equation

We take into account the bifurcation analysis of the generalized Pochhammer-Chree (PC) equation that describes the dynamics of several systems in science and engineering. The considered model is changed into a planar dynamical system by applying the Galilean transformation. The phase portraits are plotted by considering suitable values of the bifurcation parameters. The considered model is solved using the RK method to compute the supernonlinear and nonlinear wave solutions. All phase portraits and wave solutions are depicted in the phase plane by simply fixing the relevant parameters values. The equilibrium points are obtained, and the same are classified. Moreover, sensitive analysis for different initial value problems is applied to analyze the quasiperiodic, chaotic behavior and time series after introducing an extrinsic periodic perturbation term. In addition, the Lyapunov characteristic exponents, Poincare section and bifurcation diagrams are also discussed to examine the chaotic pattern of the model. Numerical simulation results show that changing the frequencies and amplitude values impacts the dynamical features of the considered model.

The null condition for systems of nonlinear wave equations and its related topics

In this article, we consider the Cauchy problem for systems of nonlinear wave equations, whose nonlinear terms depend mainly on derivatives of the unknown, with small, smooth, and compactly supported initial data. If the nonlinear terms have the critical power, we need some structural conditions to obtain the small data global existence. The null condition is one such condition, but recently some weaker conditions were also found. We discuss the small data global existence and the asymptotic behavior of solutions under these conditions. The corresponding result for nonlinear Schrödinger equations will be also discussed.

Developing Closed-Form Equations of Maximum Drag and Moment on Rigid Vegetation Stems in Fully Nonlinear Waves

Developing Closed-Form Equations of Maximum Drag and Moment on Rigid Vegetation Stems in Fully Nonlinear Waves